Asymptotic Analysis - Volume 108, issue 1-2

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ISSN 0921-7134 (P)
ISSN 1875-8576 (E)

Impact Factor 2018: 0.748

The journal Asymptotic Analysis fulfills a twofold function. It aims at publishing original mathematical results in the asymptotic theory of problems affected by the presence of small or large parameters on the one hand, and at giving specific indications of their possible applications to different fields of natural sciences on the other hand.

Asymptotic Analysis thus provides mathematicians with a concentrated source of newly acquired information which they may need in the analysis of asymptotic problems.

Abstract: We propose several continuous data assimilation (downscaling) algorithms based on feedback control for the 2D magnetohydrodynamic (MHD) equations. We show that for sufficiently large choices of the control parameter and resolution and assuming that the observed data is error-free, the solution of the controlled system converges exponentially (in L 2 and H 1 norms) to the reference solution independently of the initial data chosen for the controlled system. Furthermore, we show that a similar result holds when controls are placed only on the horizontal (or vertical) variables, or…on a single Elsässer variable, under more restrictive conditions on the control parameter and resolution. Finally, using the data assimilation system, we show the existence of abridged determining modes, nodes and volume elements.
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Abstract: In this article, we study the limit of a family of solutions to the incompressible 2D Euler equations in the exterior of a family of n k disjoint disks with centers { z i k } and radii ε k . We assume that the initial velocities u 0 k are smooth, divergence-free, tangent to the boundary and that they vanish at infinity. We allow, but we do not require, n k…→ ∞ , and we assume ε k → 0 as k → ∞ . Let γ i k be the circulation of u 0 k around the circle { | x − z i k | = ε k } . We prove that the limit as k → ∞ retains information on the circulations as a time-independent coefficient. More precisely, we assume that: (1) ω 0 k = curl u 0 k has a uniform compact support and converges weakly in L p 0 , for some p 0 > 2 , to ω 0 ∈ L c p 0 ( R 2 ) , (2) ∑ i = 1 n k γ i k δ z i k ⇀ μ weak-∗ in BM ( R 2 ) for some bounded Radon measure μ , and (3) the radii ε k are sufficiently small. Then the corresponding solutions u k converge strongly to a weak solution u of a modified Euler system in the full plane. This modified Euler system is given, in vorticity formulation, by an active scalar transport equation for the quantity ω = curl u , with initial data ω 0 , where the transporting velocity field is generated from ω , so that its curl is ω + μ . As a byproduct, we obtain a new existence result for this modified Euler system.
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